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Catharina Stroppel

Publications and source records attributed to Catharina Stroppel.

At least 19 recordsLinked to original sources
arXiv2026

Weight modules for quantum symmetric pair subalgebras

We develop a theory of weights for a quantum analogue of the symmetric pair (gl4,gl2 x gl2) realised as a quantum symmetric pair subalgebra. Based on Letzter's triangular decomposition we define Verma modules. Using magical operators that are compatible with weight spaces, we classify weight Verma modules and characterise their irreducible finite dimensional quotients. We then prove the existence of weight bases in tensor products by explicitly constructing some highest weight vectors. These constructions allow us to mimic the important aspects of the classical finite dimensional representation theory. Applications include a definition of rational representations, the BGG resolution, a Clebsch--Gordan formula, the Harish-Chandra isomorphism and central characters, as well as a classification and description of all irreducible polynomial representations.

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arXiv2025

Koszulity for semi-infinite highest weight categories

We show that any upper finite or essentially finite highest weight category where the standard objects have linear projective resolutions and the costandard objects have linear injective resolutions is Koszul. This extends the result of Agoston, Dlab, and Lukacs to the case of infinite highest weight categories. We apply this result to Khovanov algebras and representations of classical Deligne categories and show that these are Koszul.

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arXiv2025

The quantum $\epsilon$lectrical Hopf algebra and categorification of Fock space

In this article we introduce a generalization of the Khovanov--Lauda Rouquier algebras, the electric KLR algebras. These are superalgebras which connect to super Brauer algebras in the same way as ordinary KLR-algebras of type $A$ connect to symmetric group algebras. As super Brauer algebras are in Schur--Weyl duality with the periplectic Lie superalgebras, the new algebras describe morphisms between refined translation functors for this least understood family of classical Lie superalgebras with reductive even part. The electric KLR algebras are different from quiver Hecke superalgebras introduced by Kang--Kashiwara--Tsuchioka and do not categorify quantum groups. We show that they categorify a quantum version of a type $A$ electric Lie algebra. The electrical Lie algebras arose from the study of electrical networks. They recently appeared in the mathematical literature as Lie algebras of a new kind of electric Lie groups introduced by Lam and Pylyavskyy. We give a definition of a quantum electric algebra and realise it as a coideal subalgebra in some quantum group. We finally prove several categorification theorems: most prominently we use cyclotomic quotients of electric KLR algebras to categorify higher level Fock spaces.

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arXiv2024

Braiding on type A Soergel bimodules: semistrictness and naturality

We consider categories of Soergel bimodules for the symmetric groups S_n in their gl(n)-realizations for all n and assemble them into a locally linear monoidal bicategory. Chain complexes of Soergel bimodules likewise form a locally dg-monoidal bicategory which can be equipped with the structure of a braiding, whose data includes the Rouquier complexes of shuffle braids. The braiding, together with a uniqueness result, was established in an infinity-categorical setting in recent work with Yu Leon Liu, Aaron Mazel-Gee and David Reutter. In the present article, we construct this braiding explicitly and describe its requisite coherent naturality structure in a concrete dg-model for the morphism categories. To this end, we first assemble the Elias-Khovanov-Williamson diagrammatic Hecke categories as well as categories of chain complexes thereover into locally linear semistrict monoidal 2-categories. Along the way, we prove strictness results for certain standard categorical constructions, which may be of independent interest. In a second step, we provide explicit (higher) homotopies for the naturality of the braiding with respect to generating morphisms of the Elias-Khovanov-Williamson diagrammatic calculus. Rather surprisingly, we observe hereby that higher homotopies appear already for height move relations of generating morphisms. Finally, we extend the homotopy-coherent naturality data for the braiding to all chain complexes using cohomology-vanishing arguments.

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arXiv2024

Geometric categorifications of Verma modules: Grassmannian Quiver Hecke algebras

Naisse and Vaz defined an extension of KLR algebras to categorify Verma modules. We realise these algebras geometrically as convolution algebras in Borel-Moore homology. For this we introduce Grassmannian-Steinberg quiver flag varieties. They generalize Steinberg quiver flag varieties in a non-obvious way, reflecting the diagrammatics from the Naisse-Vaz construction. Using different kind of stratifications we provide geometric explanations of the rather mysterious algebraic and diagrammatic basis theorems. A geometric categorification of Verma modules was recently found in the special case of $\mathfrak{sl}_2$ by Rouquier. Rouquier's construction uses coherent sheaves on certain quasi-map spaces to flag varieties (zastavas), whereas our construction is implicitly based on perverse sheaves. Both should be seen as parts (on dual sides) of a general geometric framework for the Naisse-Vaz approach. We first treat the (substantially easier) $\mathfrak{sl}_2$ case in detail and construct as a byproduct a geometric dg-model of the nil-Hecke algebras. The extra difficulties we encounter in general require the use of more complicated Grassmannian-Steinberg quiver flag varieties. Their definition arises from combinatorially defined diagram varieties which we assign to each Naisse-Vaz basis diagram. Our explicit analysis here might shed some light on categories of coherent sheaves on more general zastava spaces studied by Feigin-Finkelberg-Kuznetsov-Mirković and Braverman, which we expect to occur in a generalization of Rouquier's construction away from $\mathfrak{sl}_2$.

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arXiv2024

Diagrammatics for the smallest quantum coideal and Jones--Wenzl projectors

We describe algebraically, diagrammatically and in terms of weight vectors, the restriction of tensor powers of the standard representation of quantum $\mathfrak{sl}_2$ to a coideal subalgebra. We realise the category as module category over the monoidal category of type $\pm1$ representations in terms of string diagrams and via generators and relations. The idempotents projecting onto the quantized eigenspaces are described as type $B/D$ analogues of Jones--Wenzl projectors. As an application we introduce and give recursive formulas for analogues of $Θ$-networks.

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arXiv2024

Khovanov algebras of type B and tensor powers of the natural $\mathrm{OSp}$-representation

We develop the theory of projective endofunctors for modules of Khovanov algebras $K$ of type B. In particular we compute the composition factors and the graded layers of the image of a simple module under such a projective functor. We then study variants of such functors for a subquotient $e\tilde{K}e$. Via a comparison of two graded lifts of the Brauer algebra we relate the Khovanov algebra to the Brauer algebra and use this to show that projective functors describe translation functors on representations of the orthosymplectic supergroup $\mathrm{OSp}(r|2n)$. As an application we get a description of the Loewy layers of indecomposable summands in tensor powers of the natural representation of $\mathrm{OSp}(r|2n)$.

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arXiv2024

A braided monoidal $(\infty,2)$-category of Soergel bimodules

The Hecke algebras for all symmetric groups taken together form a braided monoidal category that controls all quantum link invariants of type A and, by extension, the standard canon of topological quantum field theories in dimension 3 and 4. Here we provide the first categorification of this Hecke braided monoidal category, which takes the form of an $\mathbb{E}_2$-monoidal $(\infty,2)$-category whose hom-$(\infty,1)$-categories are $k$-linear, stable, idempotent-complete, and equipped with $\mathbb{Z}$-actions. This categorification is designed to control homotopy-coherent link homology theories and to-be-constructed topological quantum field theories in dimension 4 and 5. Our construction is based on chain complexes of Soergel bimodules, with monoidal structure given by parabolic induction and braiding implemented by Rouquier complexes, all modelled homotopy-coherently. This is part of a framework which allows to transfer the toolkit of the categorification literature into the realm of $\infty$-categories and higher algebra. Along the way, we develop families of factorization systems for $(\infty,n)$-categories, enriched $\infty$-categories, and $\infty$-operads, which may be of independent interest. As a service aimed at readers less familiar with homotopy-coherent mathematics, we include a brief introduction to the necessary $\infty$-categorical technology in the form of an appendix.

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arXiv2023

Existence and orthogonality of stable envelopes for bow varieties

Stable envelopes, introduced by Maulik and Okounkov, provide a family of bases for the equivariant cohomology of symplectic resolutions. The theory of stable envelopes provides a fascinating interplay between geometry, combinatorics and integrable systems. In this expository article, we give a self-contained introduction to cohomological stable envelopes of type A bow varieties (an interesting class of varieties extending Nakajima quiver varieties). Our main focus is on the existence and the orthogonality properties of stable envelopes for bow varieties. The restriction to this specific class of varieties allows us to illustrate the theory combinatorially and to provide simplified proofs, both laying a basis for explicit calculations.

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arXiv2023

Standard Extension Algebras I: Perverse Sheaves and Fukaya Calculus

In this first of a series of articles on standard extension algebras we study standard perverse sheaves on varieties with $\mathbb{G}_m$-actions. Based on Braden's hyperbolic localisation, we describe their extension algebra geometrically via a convolution structure on the intersections of attracting with repelling cells. We introduce a multiplicative structure on open Richardson varieties which provides a practical way to compose these extensions in case of flag varieties. For open Richardson varieties in Grassmannians we construct two explicit cell decompositions, of Gauss- and of Deodhar-type. It is shown that the latter is a stratification with the same combinatorics as Deodhar's decomposition. We introduce a calculus of Fukaya diagrams to encode the geometry of the decompositions. It provides a model for the cohomology of open Richardson varieties and thus for standard extensions. The calculus is motivated by the Mak-Smith Fukaya-Seidel category of a natural Lefschetz fibration and should allow to compute morphism spaces in there. We finally discuss the relation of our work to extensions of (parabolic) Verma modules in category O as well as to the computation of R- and R'-polynomials.

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arXiv2022

Categorification: tangle invariants and TQFTs

Based on different views on the Jones polynomial we review representation theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them via the theory of Soergel bimodules. The influence of these categorifications on the development of 2-representation theory and the interaction between topological invariants and 2-categorical structures is discussed. Finally, we indicate how categorified representations of quantum groups on the one hand and monoidal 2-categories of Soergel bimodules on the other hand might lead to new interesting 4-dimensional TQFTs.

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arXiv2021

Motivic Springer Theory

We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves called Springer motives. To this end, we lay foundations to a motivic Springer theory and prove formality results using weight structures. As byproduct, we express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in affine type A (with cyclic orientation) admit an affine paving.

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arXiv2021

A Lie theoretic categorification of the coloured Jones polynomial

We use the machinery of categorified Jones-Wenzl projectors to construct a categorification of a type A Reshetikhin-Turaev invariant of oriented framed tangles where each strand is labeled by an arbitrary finite-dimensional representation. As a special case, we obtain a categorification of the coloured Jones polynomial of links.

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arXiv2021

Semi-infinite highest weight categories

We develop axiomatics of highest weight categories and quasi-hereditary algebras in order to incorporate two semi-infinite situations which are in Ringel duality with each other; the underlying posets are either upper finite or lower finite. We also consider various more general sorts of stratified categories. In the upper finite cases, we give an alternative characterization of these categories in terms of based quasi-hereditary algebras and based stratified algebras, which are certain locally unital algebras possessing triangular bases.

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arXiv2020

Generic $\mathfrak{gl}_2$-foams, web and arc algebras

We define parameter dependent $\mathfrak{gl}_2$-foams and their associated web and arc algebras, and verify that they specialize to several known $\mathfrak{sl}_2$ or $\mathfrak{gl}_2$ constructions related to higher link and tangle invariants. Moreover, we show that all these specializations are equivalent, and we deduce several applications, e.g. we discuss the consequences for the associated link and tangle invariants, and their functoriality.

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